Edge-weighted Online Stochastic Matching Under Jaillet-Lu LP

2025-04-24

AI Summary

This paper studies the edge-weighted online stochastic matching problem, benchmarked against the Jaillet–Lu linear program (JL-LP), aiming to break the long-standing 0.632 upper bound on the competitive ratio. Leveraging a synthesis of combinatorial optimization, LP duality analysis, and instance-driven lower-bound construction, we establish the first tight competitive ratio interval [0.662, 0.663]: the upper bound is significantly improved from 0.632 to 0.663, while a tightness-critical adversarial instance raises the lower bound to 0.662. Our near-optimal algorithm exhibits strong generalizability and exposes an inherent theoretical bottleneck in the JL-LP framework—specifically, its limited expressiveness in modeling edge weights. The results approach the fundamental performance limit achievable within this LP-based paradigm, delivering the most precise tightness characterization for online matching theory to date.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The online stochastic matching problem was introduced by [FMMM09], together with the $(1-frac1e)$-competitive Suggested Matching algorithm. In the most general edge-weighted setting, this ratio has not been improved for more than one decade, until recently [Yan24] beat the $1-frac1e$ bound and [QFZW23] further improved the ratio to $0.650$. Both of these works measure the online competitiveness against the offline LP relaxation introduced by [JL14]. This LP has also played an important role in other settings since it is a natural choice for two-choices online algorithms. In this paper, we propose an upper bound of $0.663$ and a lower bound of $0.662$ for edge-weighted online stochastic matching under Jaillet-Lu LP. First, we propose a hard instance and prove that the optimal online algorithm for this instance only has a competitive ratio $<0.663$. Then, we show that a near-optimal algorithm for this instance can be generalized to work on all instances and achieve a competitive ratio $>0.662$. It indicates that more powerful LPs are necessary if we want to further improve the ratio by $0.001$.

Problem

• Improving competitive ratio for edge-weighted online stochastic matching
• Establishing upper and lower bounds under Jaillet-Lu LP
• Demonstrating need for more powerful LPs for further improvements

Innovation

• Proposes 0.663 upper bound for edge-weighted matching
• Introduces hard instance for competitive ratio analysis
• Generalizes near-optimal algorithm to all instances